A numerical nonisothermal two-phase wellbore model is developed to simulate downward flow of a steam and water mixture in the wellbore. This model entails simultaneous solution of coupled mass and momentum conservation equations inside the wellbore with an energy conservation equation for the fluids within the wellbore, surrounding medium, and formation. A new drift-flux model that accounts for slip between the phases inside the wellbore is employed. In addition, a two-dimensional implicit scheme that allows for heat transfer in both the axial and radial directions in the formation is developed. Furthermore, a rigorous nonlinear temperature- and depth-dependent overall heat transfer coefficient is implemented. The model predictions are validated against real field data and other available models. The model is useful for designing well completion and accurately computing the wellbore/formation heat transfer, which is very important for estimating oil recovery by using steam injection.
Modeling of steam injection wells for continuous estimation of pressure, temperature, and different phase velocities and densities as functions of depth and time is crucial for well design, steam injection projects planning and data gathering for continuous reservoir management and real time well monitoring. Once steam is injected in the well, both pressure and temperature of the injected steam and accordingly the densities of water and steam phases will change. These changes are due to the heat exchange between the steam and cold formation surrounding the well, the friction between the steam and inner tubing surface and the change of the hydrostatic pressure with respect to depth,. More importantly, the injected steam quality will drop due to the heat loss from the wellbore system towards the cold formation. The steam quality in the formation can be much worse than that in the wellhead because of an improper wellbore design, no tubular insulation, and/or the deep well location. The multiphase nature of the flow inside the wellbore, the complex heat transfer mechanisms between the wellbore and the surrounding medium, and the unsteady state nature of the flow and transport processes make the entire system intricately coupled and extremely difficult to solve.
Numerous investigators have worked on the modeling of both injection and production wells. One of the first papers goes back to Ramey(1) in 1962, which has been referred to by many subsequent works modeling the wellbore heat loss and pressure drop. In that paper, the author simplified the heat balance equation to solve it analytically. The steady-state flow of incompressible single phase, with fixed fluid and formation properties with respect to depth and temperature, was analyzed. A simple procedure was presented to couple the steady-state heat loss of the wellbore fluid with a transient heat flow in the formation via an overall heat transfer coefficient. Moreover, it was assumed that the overall heat transfer coefficient was independent of depth, and the frictional loss and kinetic energy effect were neglected. In 1965, Satter(2) improved Ramey's analytical model by considering a depth-dependent overall heat transfer coefficient and phase- and temperature-dependent fluid properties.
